I'm reading about glycolysis in a general biology textbook. It says that in the first 5 steps the free energy increases with each step. Two of those steps are powered by ATP, but the rest are not. In those other steps, how does the free energy increase? What powers it? Thanks.

Increase in free energy just refers to Gibbs free energy. That free energy comes from enthalpy (internal energy) - entropy (chaos). Also expressed as Gibbs Free Energy = Enthalpy - Entropy. The idea is that with each subsequent step of glycoysis there is more chaos and thus the entropy goes up and the free energy becomes more negative. The free energy is becoming more negative (i.e. the the magnitude of the free energy is becoming greater). Thus as the chaos increases the free energy continues to increase. The ATP usage in the two steps: hexokinase making glucose-6-phosphate and PFK1 making Fructose-1,6-Bisphosphate is made to overcome the activation energy of the subsequent step. I don't know what you understanding of biochemistry is and if that concept of free energy and activation energy needs to more thoroughly explained.

I have a fair foundation in chemical thermodynamics from my general chemistry courses, but I have not had a class in biochemistry. This question, however, is about thermodynamics. Unfortunately, while I appreciate your reponse, it doesn't answer my question. In most basic terms, for a system to move from a given energy state to a higher energy state, energy has to enter the system (because energy is conserved). In this example, where did that energy come from?

There is difference between standard Gibbs energy and actual Gibbs energy. The other reactions are close to equilibrium (thus they can be used by gluconeogenesis as well) and only the actual concentrations of the substrates/products determine which way it will go.